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Physics · dynamics

Motion and Forces

why things move (and stop)

A puck on perfectly smooth ice would slide forever. A ball in flight doesn't need a force pushing it forward. A car going twice as fast needs a four times longer braking distance. Draw forces like in your notebook, shoot hoops on the Moon, find the angle at which a block starts to slide, crash carts together, jump with a parachute and cut the string of a whirling ball.

F = m·a p = m·v F·Δt = Δp T ≤ μN
the horizontal velocity doesn't change — gravity only pulls down
Laboratorium

Three laws of motion — three experiments, plus building a free-body diagram.

First law: inertia

Third law: action and reaction

velocity A—
velocity B—
momentum A + B—
force while pushing—

Action and reaction forces are always equal and opposite, but they act on different bodies — that's why they don't cancel out. The same idea drives a rocket: Rocket →

Second law: F = m·a

a cart on a frictionless track
acceleration a0.00 m/s²
velocity v0.00 m/s
position x0.00 m
time t0.00 s

While the force acts, the velocity grows steadily — the v(t) graph is a straight line with slope a = F/m. When you let go, a = 0 and the cart keeps rolling at constant speed (first law). The slope of the x(t) graph is the velocity — that's exactly the derivative: Calculus →

Free-body diagram — build it yourself

drag an arrowhead · N = next force

Scene

ΣFx—
ΣFy—
target: m·a—
selected force—
weight normal force friction applied force rope tension net force

We draw all forces from one point — the body's center (the point-mass model). Arrowheads snap to important directions (vertical, horizontal, along the rope), and magnitudes round to half a newton.

How does it work?

A force is an interaction that can change a body's motion or deform it. We measure it in newtons (N): 1 N gives a 1 kg mass an acceleration of 1 m/s². On Earth, an apple of about 100 g weighs roughly 1 N. Mass (kg) tells you how hard it is to change a body's motion — it's the measure of its inertia.

Newton's three laws of motion

  1. The law of inertia. If the forces acting on a body balance out (net force = 0), the body stays at rest or moves in a straight line at constant speed.
  2. A net force produces acceleration in its direction — the larger the mass, the smaller the acceleration: a = Fnet/m.
  3. Action and reaction. When body A exerts a force on body B, B exerts a force of the same magnitude and opposite direction on A. These forces act on different bodies.
Fnet = m·aFg = m·gTs ≤ μsNTk = μkNg ≈ 9.81 m/s² · N — normal force (the surface's reaction) · T — friction · μs, μk — coefficients of static and kinetic friction

Friction and the inclined plane

On an incline we split the weight into a sliding component mg sin α and a pressing component mg cos α. The block stays put as long as static friction keeps up: mg sin α ≤ μsmg cos α, i.e. tan α ≤ μs. The mass cancels out — a heavy and a light block of the same material start sliding at the same angle. Once it's moving, the smaller kinetic friction acts and a = g(sin α − μk cos α).

sstop = v·tr + v²2μgreaction distance grows like v, braking distance like v²

Projectile motion

Without air, the motion splits into two independent ones: uniform horizontally (nothing pushes or slows it) and uniformly accelerated downward vertically.

x = v0 cos α · ty = v0 sin α · t − ½gt²R = v0² sin 2αgtflight = 2v0 sin αg

Momentum and impulse

Momentum p = mv is the “quantity of motion”. In a system with no external forces, the total momentum doesn't change — the law of conservation of momentum. The change in a body's momentum equals the impulse of the force: F·Δt = Δp. That's why a crumple zone, a seat belt, an airbag, a helmet and a pillow under an egg all work the same way: they stretch out the stopping time, so they reduce the force.

m1v1 + m2v2 = m1v1′ + m2v2′Favg = ΔpΔt

Air drag and circular motion

At typical speeds, air drag grows with the square of the speed. A falling body speeds up until drag matches its weight — from then on it falls at a constant terminal velocity. A body moving in a circle keeps changing the direction of its velocity, so it needs a force pointing toward the center (centripetal): string tension, tire friction, the push of the track.

Fd = ½ρCAv²Fc = mv²rvmax = √(μgr)last formula: a flat curve

Simplifications: bodies are treated as points or rigid blocks; friction is described by two constant coefficients; air drag uses the quadratic formula with a constant C (no ball spin or wind); braking at a constant deceleration μg; in the crash test we assume a constant deceleration over the given distance and a force pulse shaped like half a sine wave; the egg's breaking force is approximate. Motion is computed by our own fixed-step integrator (4th-order Runge-Kutta or semi-implicit Euler with substeps).

See also

Energy — the same motions as an energy balance Orbit — a throw that never lands Rocket — momentum and the third law Wing — forces from the air Chaos — the double pendulum Buoyancy Calculus — the derivative is velocity

Challenges

  1. Make a basket with a projectile shotIn “Projectiles” mode, turn on the game and hit the hoop (angle between 10° and 80°).not completed
  2. Find the angle at which a wooden block starts to slideWood–wood pair: increase the angle slowly until the block moves — at most 1° past the threshold.not completed
  3. Stop the car before the obstacle on a wet roadWet road surface, at least 30 km/h — the car must stop before the traffic cone.not completed
  4. Collide the carts so that both stopOn the air track, both carts must be standing still after the collision.not completed
  5. Build a correct free-body diagram for the hanging lampIn the force builder, set the tensions of both ropes so that the lamp hangs in equilibrium.not completed
  6. Skydiver: reach terminal velocityIn free fall, wait until the acceleration drops almost to zero.not completed

Did you know…?

  • In 1971 David Scott dropped a hammer and a falcon feather on the Moon — they fell at the same time, because in a vacuum all bodies fall with the same acceleration. Galileo predicted this more than 300 years earlier.
  • A car going 100 km/h hits an obstacle as hard as a fall from a height of about 39 m (h = v²/2g) — roughly from the 13th floor.
  • Astronauts on the ISS aren't weightless because there's no gravity up there. At an altitude of about 400 km, Earth's pull is still about 90% of what it is at the surface. They are simply falling all the time — together with the station, past the Earth.

Keys

Q W E R T Y
switch modes
← →
on tabs: neighboring mode; on sliders: change the value
Space
main action of the mode: push / throw / drive / collide / jump and open the parachute / cut the string
P O N
Newton: push the cart · push off · next force (arrow keys on the canvas: magnitude and direction)
S K M H
projectiles: slow motion · new hoop · shoot at the monkey · horizontal throw
C
friction: pull the block · collisions: crash test
B 1 2 3
collisions: break the balls · Newton's cradle
U
drag: drop the feather and hammer
J L
circle: car on a curve · loop